• Correlation
  • A statistical measure that describes the extent to which two variables change together
  • Does not imply causation
• Causation
  • A relationship between two variables where one variable causes the other to change
  • Establishing causation requires careful study and analysis

• Goal: Estimate the causal effect of a treatment, policy, or intervention.
• Key Question: What would have happened if things had been different?
• Example: Does a new economic policy increase employment?

• Causal effects are defined in terms of potential outcomes.
• The counterfactual is what would have happened in an alternative scenario.
• Example:
  • You take a job training program → You get a higher salary.
  • Counterfactual: What would your salary have been if you had not taken the program?

• Define treatment T:

  • T = 1 (treated)
  • T = 0 (control)

• Define potential outcomes:

  • \(Y(1)\) = Outcome if treated
  • \(Y(0)\) = Outcome if not treated

• Causal Effect:

  \[ ATE = Y(1) - Y(0) \]

• We never observe both \(Y(1)\) and \(Y(0)\) for the same unit.
• This makes causal inference challenging.
• Solution: Use design or statistical methods to approximate the counterfactual.
• The better we can approximate the counterfactual, the more accurate our causal estimates will be.

• If we want to know what would happen if a unit was treated, we can compare it to similar units that were not treated.
• Assumption: Units are similar in all respects except for the treatment.
• Challenge: How do we know if units are similar?

• Randomized Experiments: Gold standard for causal inference.
• When we are in control of the world, we can randomly assign treatment.
  • Random Assignment: Ensures that treatment is independent of potential outcomes.
  • Counterfactuals: Each unit has a well-defined counterfactual.
  • Causal Effects: Can be estimated without bias.

In a randomized controlled trial (RCT):

• Units are randomly assigned to treatment or control.
• The treatment group receives the treatment, the control group does not.
• The average treatment effect (ATE) is the difference in outcomes between the two groups.
• \(ATE = E[Y(1) - Y(0)]\)

• Ethical Concerns: Not all research questions can be answered with experiments.
• Practical Concerns: Experiments can be expensive, time-consuming, or infeasible.
• External Validity: Findings from experiments may not generalize to other contexts.

• Observational Studies: Research designs where treatment is not randomly assigned.

  • Example:
    • Treatment: Job training program
    • Outcome: Salary
    • Problem: People who choose to take the program may be different from those who don’t.

• Key Challenge: Estimating causal effects without bias.

• Using what we know about the world, we can try to approximate the counterfactual.
• Assumption: Units are similar in all respects except for the treatment.
• Methods:
  • Matching
  • Difference-in-Differences
  • Instrumental Variables
  • Regression Discontinuity.
• We’ll explore these in later sessions!

• Causal inference helps us answer “what if” questions about the world
• The potential outcomes framework defines causal effects.
• The fundamental problem of causal inference is that we never observe both potential outcomes.
• Next session: Directed Acyclic Graphs & Bias.

• Bias in [observational] studies:
  • Confounding
  • Selection Bias/Endogeneity
• DAGs help identify these sources of bias.

• Definition: Confounding occurs when a third variable (a confounder) influences both the treatment and the outcome, creating a spurious association.

• Problem: Confounders are pre-treatment variables that affect both treatment assignment and the outcome.

• Effect: Confounding distorts the causal effect because it makes it unclear whether the observed effect is due to the treatment or the confounding variable.

• Confounding is not based on statistical associations, but on qualitative knowledge of the data.

RQ: Does job training increase salary?

• Example:
  • Treatment: Job training program
  • Outcome: Salary
  • Confounder: Motivation
• Problem: The treatment and outcome are correlated because of the confounder.

• Definition: Selection bias occurs when the sample used in the analysis is not representative of the population due to a non-random selection process

• Problem: Arises when the probability of being included in the study depends on the treatment, the outcome, or factors related to both.

• Effect: Creates spurious associations between the treatment and outcome that do not reflect the true causal effect.

• Example:
  • Treatment: Job Training Program
  • Outcome: Employment
  • Selection Bias: Some people complete the training, some don’t
• Conditioning on a collider induces a spurious association between the treatment and outcome.

• Example:
  • Treatment: Going to the doctor
  • Outcome: Health
  • Selection Bias: Only sick people go to the doctor
• Conditioning on whether someone goes to the doctor induces a spurious association between going to the doctor and health.

• Survival Bias: Only observing units that survive until the end of the study.
• Non-response Bias: Only observing units that respond to the survey.
• Publication Bias: Only observing studies that report significant results.
• Attrition Bias: Only observing units that remain in the study.

• We can condition on confounding variables to block backdoor paths.
• Conditioning: Adjusting for confounders in the analysis.
  • e.g. ‘controlling for’ or ‘including’ variables in a regression model.
• We can’t really ‘control’ for all confounders, but we can try to identify and adjust for the most important ones.

• This is more challenging than confounding because it is not based on pre-treatment variables

• Solutions:

  • Randomized Experiments: Random assignment ensures that treatment is independent of potential outcomes.
  • Instrumental Variables: Use an instrument to estimate the causal effect.

• Directed Acyclic Graphs (DAGs):
  • Visual tool for representing causal relationships.
  • Help identify sources of bias in observational studies.
• Confounding:
  • A third variable influences both the treatment and outcome.
  • Distorts the causal effect.
• Selection Bias/Endogeneity:
  • Non-random selection process creates spurious associations.
  • More challenging to address than confounding.

• Resources:
  • The Effect: An Introduction to Research Design and Causality by Nick Huntington-Klein
  • (Mostly Clinical) Epidemiology with R by James Brophy
  • DAGitty: Online tool for creating and analyzing DAGs
  • ggdag: R package for visualizing DAGs

• Causal Identification: Estimating causal effects in observational studies.
• Common Methods:
  • Difference-in-Differences (DiD)
  • Regression Discontinuity (RD)
  • Instrumental Variables (IV)
• Goal: Estimate causal effects without bias.

• ATE (Average Treatment Effect):
  • The average causal effect of treatment on the outcome.
  • \(ATE = E[Y(1) - Y(0)]\)
• ATT (Average Treatment Effect on the Treated):
  • The average causal effect of treatment on the treated units.
  • \(ATT = E[Y(1) - Y(0) | T = 1]\)
• LATE (Local Average Treatment Effect):
  • The causal effect of treatment for compliers in an IV setting.
  • \(LATE = E[Y(1) - Y(0) | D = 1]\)
• CATE (Conditional Average Treatment Effect):
  • The causal effect of treatment for a specific subgroup of units.
  • \(CATE = E[Y(1) - Y(0) | X]\)

• Randomized Experiments (RCTs):
  • Gold standard for causal inference.
  • Random assignment ensures treatment is independent of potential outcomes.
  • Causal effects can be estimated without bias.
  • Estimates the ATE.
• We want to keep returning to this ideal when designing observational studies.

• Workhorse of causal inference in economics and social sciences.
• Idea: Compare changes in outcomes over time between treated and control groups.
• Estimand: The average treatment effect of the treatment on the treated.

• Assumptions:
  • Parallel Trends: Treated and control groups have parallel trends in the absence of treatment.
  • Common Shocks: No unobserved shocks that affect treated and control groups differently.
  • Stable Treatment: Treatment does not change over time.
  • No Spillovers: Treatment does not affect control units.
  • No Anticipation: Units do not anticipate the treatment.

• Treatment:
  • Local authorities that receive the training programme.
• Control:
  • Local authorities that do not receive the training programme.
• Outcome:
  • Employment rates before and after the programme is introduced.

• Assumptions:
  • Parallel Trends: Employment rates would have followed the same trend in treated and control groups in the absence of the programme.
  • Common Shocks: No unobserved shocks (other than the treatment) that affect treated and control groups differently.
  • Stable Treatment: The programme does not change over time.
  • No Spillovers: The programme does not affect control units.
  • No Anticipation: Local authorities do not anticipate the programme.

Many policies assign treatment based on a threshold. RD estimates causal effects by comparing outcomes just above and below the threshold.

• Idea: Compare outcomes for units just above and below a [arbitrary] threshold.
• Key Assumption: Units just above and below the threshold are similar in all other respects.
• Estimand: Usually the LATE (Local Average Treatment Effect).

• Assumptions:
  • Common Trend: Units just above and below the threshold would have followed the same trend in the absence of the threshold.
  • No Manipulation: Units cannot manipulate the threshold to change treatment assignment.
  • No Spillovers: Treatment does not affect units on the other side of the threshold.

• Treatment:
  • Seminar groups with more than 30 students.
• Control:
  • Seminar groups with 30 or fewer students.
• Outcome:
  • Student performance just above and below the threshold.

• Assumptions:
  • Common Trend: Student performance would have followed the same trend in large and small seminar groups in the absence of the threshold.
  • No Manipulation: Students cannot manipulate seminar group size to change treatment assignment.
  • No Spillovers: Seminar group size does not affect students in other groups.

• Assumptions:
  • Relevance: The instrument affects treatment.
  • Exogeneity: The instrument is unrelated to the outcome except through the treatment.
  • Exclusion: The instrument does not affect the outcome except through the treatment.

• Treatment:
  • College attendance.
• Instrument:
  • Proximity to a college.
• Outcome:
  • Earnings.

• Assumptions:
  • Relevance: Proximity to a college affects college attendance.
  • Exogeneity: Proximity to a college is unrelated to earnings except through college attendance.
  • Exclusion: Proximity to a college does not affect earnings except through college attendance.

• How might proximity to a college be related to earnings?
  • How might being close to a college affect earnings through other channels?
  • How might the exclusion restriction be violated?

• Causal Identification: Estimating causal effects in observational studies.
• Methods:
  • Difference-in-Differences (DiD): Compare changes in outcomes over time between treated and control groups.
  • Regression Discontinuity (RD): Compare outcomes just above and below a threshold.
  • Instrumental Variables (IV): Use an instrument to estimate the causal effect.
• Assumptions: Each method relies on specific assumptions to identify causal effects.

• Idea: Compare changes in outcomes over time between treated and control groups.
• Assumptions:
  • Parallel Trends: Treated and control groups have parallel trends in the absence of treatment.
  • Common Shocks: No unobserved shocks that affect treated and control groups differently.
  • Stable Treatment: Treatment does not change over time.
  • No Spillovers: Treatment does not affect control units.
  • No Anticipation: Units do not anticipate the treatment.

• Treatment:
  • States that Trump targeted.
• Control:
  • States that were not targeted.
• Outcome:
  • Social distancing behavior before and after the tweets.

• Parallel Trends: Social distancing behavior would have followed the same trend in treated and control states in the absence of the tweets.
• Common Shocks: No unobserved shocks (other than the tweets) that affect treated and control states differently.
• No Spillovers: The tweets do not affect control states.
• No Anticipation: States do not anticipate the tweets.

• Common Shocks: What other events might have affected social distancing behavior?
• No Spillovers: How might the tweets affect control states?

• \(Y_{it}\) = Mobility for state \(i\) on day \(t\)
• \(\alpha_i\) = state fixed effects
• \(lambda_t\) = day fixed effects
• \(\delta\) = estimated causal effect of the treatment
• \(\text{treated}_i\) = indicator for treated states
• \(\text{post}_t\) = indicator for post-treatment period
• \(\epsilon_{it}\) = error term

• Difference-in-Differences (DiD):
  • Compare changes in outcomes over time between treated and control groups.
  • Assumptions: Parallel Trends, Common Shocks, Stable Treatment, No Spillovers, No Anticipation.
  • Estimating the causal effect: DiD estimator.

• Potential Outcomes Framework:
  • Defines causal effects in terms of potential outcomes.
  • The fundamental problem of causal inference.
• Directed Acyclic Graphs (DAGs):
  • Visual tool for representing causal relationships.
  • Identify sources of bias in observational studies.
• Causal Identification in Observational Studies:
  • Methods for estimating causal effects without bias.
  • Difference-in-Differences, Regression Discontinuity, Instrumental Variables.
• Estimating Causal Effects in Observational Studies:
  • A lot more challenging than in experiments.
  • Requires careful attention to assumptions and methods.